Question

How do you write $ {27^{\dfrac{1}{3}}} $ in radical form?

Answer 1

Hint: In this question we need to write $ {27^{\dfrac{1}{3}}} $ in radical form. Then we will use a law of radical, to write the given term into the simplest form of radical. The law states that, if $ n $ is a positive integer that is greater than $ x $ and $ a $ is a real number or a factor, then $ {a^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}} $ . Then we will substitute the values and determine the radical form of $ {27^{\dfrac{1}{3}}} $ .

Complete step-by-step answer:
Here, we need to write $ {27^{\dfrac{1}{3}}} $ in radical form.
Expressing in simplest radical form is nothing but simplifying the radical into the simplest form with no more square roots, cube roots, etc. left to find. In other words, a number under a radical is indivisible by a perfect square other than $ 1 $ .
If $ n $ is a positive integer that is greater than $ x $ and $ a $ is a real number or a factor, then
 $ {a^{\dfrac{x}{n}}} = \sqrt[n]{{{a^x}}} $
It means raise $ a $ to the power $ x $ then find the $ {n^{th}} $ root of the result.
Here, $ a = 27 $ , $ x = 1 $ and $ n = 3 $ .
Now, let us substitute the values in the equation, we have,
 $ {27^{\dfrac{1}{3}}} = \sqrt[3]{{27}} $
 $ {27^{\dfrac{1}{3}}} = 3 $ .
Hence, the required answer is $ 3 $ .
So, the correct answer is “3”.

Note: In this question it is important to note here that we used a law of the radical to solve this form i.e., a radical represents a fractional exponent in which the numerator of the fractional exponent is the power of the base and the denominator of the fractional exponent is the index of the radical. However, if it is not mentioned as a radical form in the given question, then we can solve it by rewriting $ 27 $ as $ {3^3} $ . Then we will have $ {\left( {{3^3}} \right)^{\dfrac{1}{3}}} $ , here we know from the law of exponents that $ {\left( {{a^m}} \right)^n} = {a^{m \times n}} $ , then we have $ {3^{\dfrac{3}{3}}} $ where we can cancel out the $ 3 $ at numerator and denominator. By which we will get the answer as $ 3 $ .